Convergence rate of collocation method based on wavelet for nonlinear weakly singular partial integro-differential equation arising from viscoelasticity

Abstract

The main aim of this research article is to propose and analyze a Legendre wavelet collocation method (LWCM) for the nonlinear weakly singular partial integro-differential equation (SPIDE) arising from viscoelasticity subject to the given initial and boundary conditions. This problem can be found in the mathematical modeling of physical phenomena involving viscoelastic forces. Operational matrix of integration of Legendre wavelets along with collocation method are utilized to reduce the original SPIDE into the nonlinear system of algebraic equations. Some numerical results are presented to simplify applications of operational matrix formulation and reduce the computational cost. Convergence analysis, numerical stability and rate of convergence (C-order) of the proposed method are also investigated by considering a test function. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the L2 and L∞ norms. Finally, we compare the proposed LWCM with well-known Crank-Nicolson and Crandall's methods (for instance, see Table 4). © 2018 Wiley Periodicals, Inc.